#7117813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,7)(3,8,4,6)(10,13)(11,15)(17,18,23,24,22,26,21,20)(19,27,30,32,28,25,31,29);(10,13)(11,15)(18,24)(19,28)(20,26)(21,23)(25,32)(27,29);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,32)(21,31)(22,28)(23,30)(24,29)(26,27);(10,13)(14,16)(17,20,23,18,22,24,21,26)(19,29,30,27,28,32,31,25);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,24,26,20)(19,30,28,31)(25,32,27,29)

Polycyclic presentation:

2^9|a^2=fghi,c^2=gh,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251178, #72251190, #72251252, …
maximal normal subgroup of	#72251178, #72251190, #72251252, …
maximal quotient of	#72264667

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7025009);